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I have two Poisson processes A and B having arrival rates $$\lambda_{1}$$ and $$\lambda_{2}$$.

What is the probability that only 1 arrival appears from Process A and 0 arrival occurs from process B in the time interval [0-10s]? I mean what is the probability that in 10s time only 1 arrival from Process A appears?

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2 Answers 2

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Well, $~\mathsf P(A_{(0;10]}{=}1 \cap B_{(0;10]}{=}0) ~=~ \mathsf P(A_{(0;10]}{=}1)\,\mathsf P(B_{(0;10]}{=}0)~$ if they are independent processes, which you neglected to mention if it is so or not.

If so, it is just a case of applying the Poisson Distribution's probability mass function for the interval and rates.

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  • $\begingroup$ Thank you @Graham Kemp. Yes they are independent. You are right. I forgot to mention in my question. $\endgroup$ Commented Jul 18, 2017 at 16:42
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here mean number of events per interval = m Process 1 mean arrival rate = m1 process 2 mean arriva rate = m2

P(A)= probability that there is an arrival from process1 P(B)= probability that there is no arrival from proecss 2 p(A|B)= P(A) as p(A^B) = P(A)P(B)/P(B)= P(A) thus, P(X=1)=(E^(-1)*1)/1= 1/e

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  • $\begingroup$ thank you very much for your answer. I understand what you mean. However, my question is then, in my case then what is the difference between a simple poisson process and a merged poisson process. Although there are independent, but still in the time interval of 10 s, is there no impact of process B on the arrival probability of process A $\endgroup$ Commented Jul 18, 2017 at 16:36

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